We study a nonlinear Langevin equation involving a Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function having a singularity. Under a regularity assumption of solution of the problem, we show that the problem can be transformed to a Volterra integral equation with two parameters Mittag-Leffler function in the kernel. Base on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry-Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.
{"title":"Existence and continuity results for a nonlinear fractional Langevin equation with a weakly singular source","authors":"Nguyen Minh Dien","doi":"10.1216/jie.2021.33.349","DOIUrl":"https://doi.org/10.1216/jie.2021.33.349","url":null,"abstract":"We study a nonlinear Langevin equation involving a Caputo fractional derivatives of a function with respect to another function in a Banach space. Unlike previous papers, we assume the source function having a singularity. Under a regularity assumption of solution of the problem, we show that the problem can be transformed to a Volterra integral equation with two parameters Mittag-Leffler function in the kernel. Base on the obtained Volterra integral equation, we investigate the existence and uniqueness of the mild solution of the problem. Moreover, we show that the mild solution of the problem is dependent continuously on the inputs: initial data, fractional orders, appropriate function, and friction constant. Meanwhile, a new Henry-Gronwall type inequality is established to prove the main results of the paper. Examples illustrating our results are also presented.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41521159","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper Tricomi-Gellerstedt-Keldysh-type fractional elliptic equations are studied. The results on the well-posedness of fractional elliptic boundary value problems are obtained for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains.
{"title":"Well-posedness of Tricomi–Gellerstedt–Keldysh-type fractional elliptic problems","authors":"Michael Ruzhansky, B. Torebek, B. Turmetov","doi":"10.1216/jie.2022.34.373","DOIUrl":"https://doi.org/10.1216/jie.2022.34.373","url":null,"abstract":"In this paper Tricomi-Gellerstedt-Keldysh-type fractional elliptic equations are studied. The results on the well-posedness of fractional elliptic boundary value problems are obtained for general positive operators with discrete spectrum and for Fourier multipliers with positive symbols. As examples, we discuss results in half-cylinder, star-shaped graph, half-space and other domains.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44586168","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Existence of a solution of the functional integral equation in an unbounded interval involving the Riemann–Liouville operator is investigated. Here sufficient conditions in the context of existence and stability are derived by employing hybridized fixed point theory in the Banach algebra setting. Further, an example is presented to showcase the validity of the obtained result. Moreover, the solution of the example in closed form is estimated by the semianalytic technique which is being driven by a modified homotopy perturbation method in conjunction with the Adomian decomposition method.
{"title":"Existence criteria and solution search by the analytic technique of functional integral equation","authors":"Dipankar Saha, M. Sen","doi":"10.1216/jie.2021.33.247","DOIUrl":"https://doi.org/10.1216/jie.2021.33.247","url":null,"abstract":"Existence of a solution of the functional integral equation in an unbounded interval involving the Riemann–Liouville operator is investigated. Here sufficient conditions in the context of existence and stability are derived by employing hybridized fixed point theory in the Banach algebra setting. Further, an example is presented to showcase the validity of the obtained result. Moreover, the solution of the example in closed form is estimated by the semianalytic technique which is being driven by a modified homotopy perturbation method in conjunction with the Adomian decomposition method.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45300211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A method of solving a nonlinear boundary value problem for the\u0000 Fredholm integro-differential equation","authors":"D. Dzhumabaev, S. Mynbayeva","doi":"10.1216/JIE.2021.33.53","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.53","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"33 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41785464","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Numerical analysis of asymptotically convolution evolutionary\u0000 integral equations","authors":"E. Messina, A. Vecchio","doi":"10.1216/JIE.2021.33.91","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.91","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47822008","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Stability and approximation of almost automorphic solutions on\u0000 time scales for the stochastic Nicholson's blowflies model","authors":"Soniya Dhama, S Z Abbas, R. Sakthivel","doi":"10.1216/JIE.2021.33.31","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.31","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46187023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Solvability for generalized nonlinear functional integral\u0000 equations in Banach spaces with applications","authors":"A. Deep, Deepmala, J. Roshan","doi":"10.1216/JIE.2021.33.19","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.19","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42961521","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On quasinormality of singular integral operators with Cauchy\u0000 kernel on $L^{2}$","authors":"E. Ko, J. Lee","doi":"10.1216/JIE.2021.33.77","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.77","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45925338","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Local existence and global nonexistence of a solution for a\u0000 Love equation with infinite memory","authors":"K. Zennir, Tosiya Miyasita, P. Papadopoulos","doi":"10.1216/JIE.2021.33.117","DOIUrl":"https://doi.org/10.1216/JIE.2021.33.117","url":null,"abstract":"","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46312791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.
{"title":"Asymptotic expansion of iterated Galerkin solution of Fredholm integral equations of the second kind with Green's kernel","authors":"Gobinda Rakshit, Akshay S. Rane","doi":"10.1216/jie.2020.32.495","DOIUrl":"https://doi.org/10.1216/jie.2020.32.495","url":null,"abstract":"We consider a Fredholm integral equation of the second kind with kernel of the type of Green’s function. Iterated Galerkin method is applied to such an integral equation. For r≥1, a space of piecewise polynomials of degree ≤r−1 with respect to a uniform partition is chosen to be the approximating space. We obtain an asymptotic expansion for the iterated Galerkin solution at the partition points. Richardson extrapolation is used to increase the order of convergence. A numerical example is considered to illustrate our theoretical results.","PeriodicalId":50176,"journal":{"name":"Journal of Integral Equations and Applications","volume":"32 1","pages":"495-507"},"PeriodicalIF":0.8,"publicationDate":"2020-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49196747","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：